A three-dimensional micropolar beam model with application to the finite deformation analysis of hard-magnetic soft beams

Abstract

The main purpose of this contribution is to develop a three-dimensional (3D) nonlinear beam model based on the micropolar continuum theory. To do so, a kinematic model based on the deformation of three directors and accounting for the micro-rotation tensor of the micropolar theory is introduced. One of the main characteristics of the present beam model is that 3D constitutive equations without any modification can be directly used in the formulation. Furthermore, it is known that a body couple field is induced in hard-magnetic soft materials (HMSMs) when subjected to external magnetic fluxes. Therefore, the stress tensor in HMSMs is asymmetric, in general. Since the asymmetry of stress is one of the main features of the micropolar theory, the present formulation can be used for analyzing the deformation of beams made of HMSMs. Accordingly, the virtual external work of the present model is formulated so that it accounts for the contribution from uniform or constant-gradient external magnetic fluxes on the beam. Moreover, a Total Lagrangian (TL) nonlinear finite element (FE) formulation to provide numerical solutions of the related problems is developed. Several numerical examples are solved to investigate the capability of the developed formulation. It is shown that the present formulation can model the size-dependent behavior of beam-like structures if the material length-scale parameter of the micropolar constitutive model is comparable to the thickness of the beam. Moreover, the proposed model can successfully predict the finite deformation of 3D beams made of HMSMs subjected to magnetic loading.